Selected topics in quantum scattering theory and emerging optimal-transport methods

Quantum scattering connects microscopic dynamics to amplitudes, phase shifts, cross sections, transport coefficients, and resonances. This selective Review organizes the Lippmann–Schwinger equation, resolvents, transfer and scattering matrices, nonequilibrium Green’s functions, and R-matrix theory as related representations of a common scattering problem. Born, Wentzel–Kramers–Brillouin, and eikonal error tests accompany comparisons of numerical scaling, stability, and method selection. Inverse scattering is examined through the effects of phase loss, incomplete aperture, Jacobian null spaces, singular-value decay, model discrepancy, and noise, together with regularized, Bayesian, and learned approaches for stabilizing reconstruction. The Review then discusses probability-level optimal transport and its extension to emerging amplitude-based formulations, using the Quantum Optimal Transport Amplitude Method (QOTAM) as a concrete example. Its Euclidean and real-time variational constructions are summarized and tested across resonant barriers, multichannel and angular scattering, three-dimensional targets, and disordered waveguides. Independent reference calculations verify complex-amplitude closure, unitarity, flux conservation, and optical-theorem relations to benchmark-dependent numerical precision. In inverse problems, QOTAM is found to be competitive with correctly specified phase-sensitive likelihood methods while showing useful robustness in selected model-mismatch tests. These examples illustrate the promise of optimal-transport-based scattering methods. More broadly, the Review emphasizes how physical constraints, uncertainty quantification, identifiability, and computational scalability jointly shape reliable forward and inverse-scattering analyses.